Problem 14 - Elementary Algebra

Question

Write each of the following so that only positive exponents appear. $$ \frac{3 a(a-5 b)^{-2}}{5 b(a-4 b)^{5}} $$

Step-by-Step Solution

Answer

Question: Rewrite the given expression with only positive exponents: $\frac{3a(a-5b)^{-2}}{5b(a-4b)^{5}}$ Answer: $\frac{3a}{5b(a-4b)^{5}(a-5b)^2}$

Step 1: Rewrite the negative exponents

First, let's identify the terms with negative exponents. In the given expression, we have \((a-5b)^{-2}\). To rewrite this term with a positive exponent, we can use the property \(a^{-n} = \frac{1}{a^n}\). Hence, we can rewrite \((a-5b)^{-2}\) as \(\frac{1}{(a-5b)^2}\). Now, let's substitute this into the original expression: $$ \frac{3a(a-5b)^{-2}}{5b(a-4b)^{5}} = \frac{3a}{5b(a-4b)^{5}} \times \frac{1}{(a-5b)^2} $$

Step 2: Multiply the fractions

By multiplying the two fractions, we can simplify the expression: $$ \frac{3a}{5b(a-4b)^{5}} \times \frac{1}{(a-5b)^2} = \frac{3a}{5b(a-4b)^{5}(a-5b)^2} $$

Step 3: Final answer

Now the expression has only positive exponents, and the final answer is: $$ \frac{3a}{5b(a-4b)^{5}(a-5b)^2} $$

Key Concepts

Understanding Negative Exponents

When dealing with exponents in algebra, you might encounter expressions with negative exponents, like \( (a-5b)^{-2} \). Understanding how to work with negative exponents is crucial. The rule for negative exponents is simple: an expression with a negative exponent can be transformed into a positive exponent by taking the reciprocal of the base. Essentially, \( a^{-n} = \frac{1}{a^n} \).

So, by applying this rule, you would convert \( (a-5b)^{-2} \) into \( \frac{1}{(a-5b)^2} \). This step is vital because it sets the stage for further simplification and ensures that all exponents in the final expression are positive.

Simplified Demonstration:

• Original: \( (x)^{-3} \)
• Converted: \( \frac{1}{(x)^3} \)

Simplifying Algebraic Expressions

Simplifying algebraic expressions requires several steps, including distributing, combining like terms, and applying exponent rules. The goal is to make the expression as straightforward as possible. In our exercise, simplification involves combining the components of the algebraic fraction after dealing with negative exponents.

Here, you multiply the fractions straight across, both numerators and denominators separately. By multiplying the given fractions \( \frac{3a}{5b(a-4b)^5} \) and \( \frac{1}{(a-5b)^2} \), you obtain \( \frac{3a}{5b(a-4b)^{5}(a-5b)^2} \). This expression is simpler because there are no longer any negative exponents, and all like terms are already combined.

Example of Simplification:

• Before: \( \frac{a \times a^{-1}}{b} \)
• After: \( \frac{1}{b} \)

Working with Algebraic Fractions

Algebraic fractions can be intimidating, but they follow the same principles as numerical fractions. In our exercise, after dealing with negative exponents, we're left with an algebraic fraction that needs to be simplified. The key to simplifying algebraic fractions is to focus on factoring and canceling out common terms when possible. However, our simplified expression \( \frac{3a}{5b(a-4b)^{5}(a-5b)^2} \) does not have common factors that can be canceled in this case.

Thus, the simplified form retains all terms in the denominator. Remember, the goal is to present a rational expression where all exponents are positive and the expression itself is as clear and simplified as possible, which facilitates further operations or evaluations if necessary.

Tips for Algebraic Fractions:

• Factor completely to identify common terms.
• Cancel out common factors in the numerator and denominator.
• Do not cancel out terms that are not common factors.